Modelling stock price movements: multifractality or multifractionality?

Abstract

The scaling properties of two alternative fractal models recently proposed to characterize the dynamics of stock market prices are compared. The former is the Multifractal Model of Asset Return (MMAR) introduced in 1997 by Mandelbrot, Calvet and Fisher in three companion papers. The latter is the multifractional Brownian motion (mBm), defined in 1995 by Péltier and Lévy Véhel as an extension of the very well-known fractional Brownian motion (fBm).

 We argue that, when fitted on financial time series, the partition function as well as the scaling function of the mBm, i.e. of a generally non-multifractal process, behave as those of a genuine multifractal process. The analysis, which concerns the daily closing prices of eight major stock indexes, suggests to evaluate prudently the recent findings about the multifractal behaviour in finance and economics.

Keywords:

• Multifractals
• MMAR
• Multifractionality
• Stock prices

Acknowledgments

The authors wish to thank the anonymous referees whose remarks have significantly improved the quality of this paper.

Notes

†Shortly, it is called ‘1/f noise’, the type of noise whose power spectra P(f) as a function of the frequency f behaves like P(f) = f ^−α, for some positive real α.

†The notion of multifractal measure substantially founds the whole literature related to multifractal analysis. For an introduction and motivation see, e.g. Mandelbrot (1999) or Harte (2001).

‡Expressed briefly, Mandelbrot and Taylor (1967) proved that a sequence with stable increments having a characteristic exponent α less than 2 can be written as a subordinated process with normal increments, the variance process of which follows a stable distribution of its own with characteristic exponent α-1 (meaning that the expected value of the variance process is infinite, or undefined).

†Given the metric spaces (X,d_X ) and (Y,d_Y), the function f:X→Y is called a Hölder function with exponent β >0 if, for each x,y∈X such that there exists a constant k satisfying the condition

The hölderianity is sufficient (but not necessary) for the continuity for the process.

‡The representation of the mBm given by Benassi et al. (1997) is of “harmonizable” type and differs from the one considered here, which is of “moving-average” type. The authors in fact assume the following: Definition. Let be a function in where . The multifractional Brownian motion of order α(x) is defined by

where dW(ξ) is the white noise.

§See Ayache (2000) for a discussion on how to generalize the mBm in order to obtain possibly multifractal scaling.

†The simulation has been carried out by using the circulant embadding method (Wood and Chan 1994).

† The following normalization has been considered: